Articles of connections

Covariant derivative in $\mathbb{R}^n$

I am studying my lecture notes on covariant derivative, and is having difficulty to do a computation: Suppose $X,Y$ be smooth vector fields in $\mathbb{R}^n.$ Consider the integral curve $c_p:(-\epsilon,\epsilon) \to \mathbb{R}^n$ of $X$ passing through $p\in \mathbb{R}^n$ at time $t=0$. We have an identification $T_{c_p(t)}\mathbb{R}^n \to T_p\mathbb{R}^n$ of the tangent spaces. Using this, we […]

Connections and Ricci identity

Given $\nabla$ a torsionless connection, the Ricci identity for co-vectors is $$\nabla_a \nabla_b \lambda_c – \nabla_b \nabla_a \lambda_c = -R^d_{\,\,cab}\lambda_d.$$ Prove $R^a_{[bcd]} = 0$ by considering the co-vector field $\lambda_c = \nabla_c f$. By definition, $$R^a_{[bcd]} = 0 = \frac{1}{3!} \left(R^a_{\,\,bcd} + R^a_{\,\,cdb} + R^a_{\,\,dbc} – R^a_{\,\,bdc} – R^a_{\,\,cbd} – R^a_{\,\,dcb}\right)\,\,\,\,(1)$$ Attempt: Input the given […]

Parallel transport along a 2-sphere.

I’m currently learning about parallel transport and connections and we were considering the parallel transport of a tangent vector along a sphere as given in the picture below. From my understanding, by defining a connection on your manifold, you provide a way to identify vectors at one point of the manifold with vectors at another […]

Equivalence for Christoffel symbol and Koszul formula

I am trying to show to define a Levi-civita connection, it’s equivalent to define Christoffel symbols or define Koszul formula. $$ 2g(\nabla_XY, Z) = \partial_X (g(Y,Z)) + \partial_Y (g(X,Z)) – \partial_Z (g(X,Y))+ g([X,Y],Z) – g([X,Z],Y) – g([Y,Z],X)$$ One direction is easy, given Koszul formula, take $X=\frac{\partial}{\partial x^i},Y=\frac{\partial}{\partial x^j}, Z=\frac{\partial}{\partial x^k}$ and compute. For the other […]

Parallel vector fields imply a flat connection?

Let $M$ be a Riemann surface and let $\nabla$ be its Levi-Cevita connection. In particular, $\nabla$ is torsion free, i.e. $\nabla_X Y – \nabla_Y X = [X,Y]$ for vector fields $X$ and $Y$. Question: Suppose there exist linearly independent vector fields $X$ and $Y$ on $M$ such that $\nabla_XY = \nabla_YX = 0$, i.e. each […]

Splitting of the tangent bundle of a vector bundle and connections

Let $\pi:E\to M$ be a smooth vector bundle. Then we have the following exact sequence of vector bundles over $E$: $$ 0\to VE\xrightarrow{} TE\xrightarrow{\mathrm{d}\pi}\pi^*TM\to 0 $$ Here $VE$ is the vertical bundle, that is kernel of the bunble map $\mathrm{d}\pi$. This sequence splits, so there is a bundle morphism $\sigma:\pi^*TM\to TE$ such that $\mathrm{d}\pi\circ\sigma=\mathrm{Id}$. It […]

Recovering connection from parallel transport

(doCarmo, Riemannian Geometry, p.56, Q2) I want to prove that the Levi-Civita connection $\nabla$ is given by $$ (\nabla_X Y)(p) = \frac{d}{dt} \Big(P_{c,t_0,t}^{-1}(Y(c(t)) \Big) \Big|_{t=t_0}, $$ where $p \in M$, $c \colon I \to M$ is an integral curve of $X$ through $p$, and $P_{c,t_0,t} \colon T_{c(t_0)}M \to T_{c(t)}M$ is the parallel transport along $c$, […]

Given a diffeomorphism between two surfaces, is there an expression for the pullback of the covariant derivative of a vector field?

Let $A$ and $B$ be two surfaces (smooth enough) in an affine space $M$ with metric $g$. Let $g^A$, $g^B$ be the metric tensors on the two surfaces induced by $g$, and $\nabla^A$, $\nabla^B$ the Levi-Civita connections on the two surfaces. Let $f:A\rightarrow B$ be a diffeomorphism. I’m wondering if, given a vector field $X^B$ […]

Covariant derivative versus exterior derivative

$\def\alt{\textrm{Alt}} \def\d{\mathrm{d}} \def\sgn{\mathrm{sgn}\,}$Let $\nabla$ be a symmetric linear connection in $M$, and $\omega$ be a $k$-form in $M$. I’m trying to find a relation between $\d\omega$ and $\nabla \omega$. I follow Spivak’s notation in Calculus on Manifolds and write $$\alt(\nabla \omega)(X_1,\cdots,X_{k+1}) = \frac{1}{(k+1)!}\sum_{\sigma \in S_{k+1}} (\sgn \sigma)\nabla\omega(X_{\sigma(1)},\cdots,X_{\sigma(k+1)}).$$I also assume the formula $$\d\omega(X_1,\cdots,X_{k+1})=\sum_{i=1}^k(-1)^{i+1}X_i(\omega(X_1,\cdots,\widehat{X_i},\cdots,X_{k+1})) + \sum_{i<j}(-1)^{i+j}\omega([X_i,X_j],X_1,\cdots,\widehat{X_i},\cdots,\widehat{X_j},\cdots,X_{k+1}).$$I’d expect […]

A diffeomorphism which maps geodesics to geodesics preserves the connection?

Let $(M,\nabla^M),(N,\nabla^N)$ be two smooth manifolds with given (affine) connections on their (tangent bundles). We say a diffeomorphism ,$\phi:(M,\nabla^M)\rightarrow(N,\nabla^N)$ is an isomorphism if: $\nabla^N_X{Y}=\phi_* \left( \nabla^{M}_{\phi^{-1}_*(X)} {\phi^{-1}_*(Y)} \right) \forall X,Y \in \Gamma(TN)$, where the pushforward $\phi_*(X)(q)=d\phi_{\phi^{-1}(q)}[X \left(\phi^{-1}(q)\right)]$ is the corresponding isomoprhism of Lie algebras. Assume $(M,\nabla)$ is a smooth manifold with an affine connection (on […]